Best Known (34, 34+10, s)-Nets in Base 32
(34, 34+10, 210212)-Net over F32 — Constructive and digital
Digital (34, 44, 210212)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 496)-net over F32, using
- net defined by OOA [i] based on linear OOA(327, 496, F32, 5, 5) (dual of [(496, 5), 2473, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(327, 993, F32, 5) (dual of [993, 986, 6]-code), using
- net defined by OOA [i] based on linear OOA(327, 496, F32, 5, 5) (dual of [(496, 5), 2473, 6]-NRT-code), using
- digital (27, 37, 209716)-net over F32, using
- net defined by OOA [i] based on linear OOA(3237, 209716, F32, 10, 10) (dual of [(209716, 10), 2097123, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3237, 1048580, F32, 10) (dual of [1048580, 1048543, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(3237, 1048576, F32, 10) (dual of [1048576, 1048539, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3233, 1048576, F32, 9) (dual of [1048576, 1048543, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(320, 4, F32, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(3237, 1048580, F32, 10) (dual of [1048580, 1048543, 11]-code), using
- net defined by OOA [i] based on linear OOA(3237, 209716, F32, 10, 10) (dual of [(209716, 10), 2097123, 11]-NRT-code), using
- digital (2, 7, 496)-net over F32, using
(34, 34+10, 419433)-Net in Base 32 — Constructive
(34, 44, 419433)-net in base 32, using
- net defined by OOA [i] based on OOA(3244, 419433, S32, 10, 10), using
- OA 5-folding and stacking [i] based on OA(3244, 2097165, S32, 10), using
- discarding factors based on OA(3244, 2097167, S32, 10), using
- discarding parts of the base [i] based on linear OA(12831, 2097167, F128, 10) (dual of [2097167, 2097136, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1283, 15, F128, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,128) or 15-cap in PG(2,128)), using
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- Reed–Solomon code RS(125,128) [i]
- discarding factors / shortening the dual code based on linear OA(1283, 128, F128, 3) (dual of [128, 125, 4]-code or 128-arc in PG(2,128) or 128-cap in PG(2,128)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12831, 2097167, F128, 10) (dual of [2097167, 2097136, 11]-code), using
- discarding factors based on OA(3244, 2097167, S32, 10), using
- OA 5-folding and stacking [i] based on OA(3244, 2097165, S32, 10), using
(34, 34+10, 3054228)-Net over F32 — Digital
Digital (34, 44, 3054228)-net over F32, using
(34, 34+10, large)-Net in Base 32 — Upper bound on s
There is no (34, 44, large)-net in base 32, because
- 8 times m-reduction [i] would yield (34, 36, large)-net in base 32, but